# On smooth foliations with Morse singularities

## TL;DR

This paper generalizes classical results on smooth foliations with Morse singularities, classifying possible singularity arrangements and extending key theorems like Haefliger and Novikov to singular settings.

## Contribution

It describes possible configurations of singularities in Morse foliations and extends classical theorems to cases with Morse singularities.

## Key findings

- Classifies arrangements of Morse singularities in foliations.
- Provides an elimination procedure for certain singularity configurations.
- Extends Haefliger and Novikov theorems to singular foliations with weakly stable components.

## Abstract

Let $M$ be a smooth manifold and let $\F$ be a codimension one, $C^\infty$ foliation on $M$, with isolated singularities of Morse type. The study and classification of pairs $(M,\F)$ is a challenging (and difficult) problem. In this setting, a classical result due to Reeb \cite{Reeb} states that a manifold admitting a foliation with exactly two center-type singularities is a sphere. In particular this is true if the foliation is given by a function. Along these lines a result due to Eells and Kuiper \cite{Ku-Ee} classify manifolds having a real-valued function admitting exactly three non-degenerate singular points. In the present paper, we prove a generalization of the above mentioned results. To do this, we first describe the possible arrangements of pairs of singularities and the corresponding codimension one invariant sets, and then we give an elimination procedure for suitable center-saddle and some saddle-saddle configurations (of consecutive indices). In the second part, we investigate if other classical results, such as Haefliger and Novikov (Compact Leaf) theorems, proved for regular foliations, still hold true in presence of singularities. At this purpose, in the singular set, $Sing(\F)$ of the foliation $\F$, we consider {\em{weakly stable}} components, that we define as those components admitting a neighborhood where all leaves are compact. If $Sing(\F)$ admits only weakly stable components, given by smoothly embedded curves diffeomorphic to $S^1$, we are able to extend Haefliger's theorem. Finally, the existence of a closed curve, transverse to the foliation, leads us to state a Novikov-type result.